Controllability of quantum systems having weakly conically connected spectrum

We explore the controllability of a closed multi-input control-affine quantum system. Previous studies have demonstrated that a spectrum connected by conical intersections which do not pile up yields exact controllability in finite dimension and approximate controllability in infinite dimension. Actually, the property that intersections between eigenvalues are conical and that they do not pile up is generic. However, in physical situations, due to symmetry of the system, the spectrum can exhibit intersections that are not conical and possibly pile up. We extend the controllability result to cover this type of situations under the hypothesis that the intersections have at least one conical direction and the piled-up intersections have "rationally unrelated germs". Finally, we provide a testable first-order sufficient condition for controllability. Physically relevant examples are provided